
0-cells (points):
This level is actually put aside by spinfoam models, which focus on geometrical structures with co-dimensions less than or equal to 2. This allows for conical singularities \(\textit {e.g.}\)  [1]}, which can be appropriately controlled in sums over random discrete geometries \(\textit {e.g.}\)  [2]}.

1-cells (edges):
Edges have the topology of a basic segment, devoid of any winding information.
One dresses edges with irreducible representations of the Lie group \(\mathrm {SU}(2)\) . These are labelled by half-integers \(j\in 2\) , usually referred to as spins. The Hilbert space \({\mathcal {V}}_{j}\)  carrying the representation of spin \(j\)  is of dimension \(d_{j}=(2j+1)\)  and its standard basis is given the spin basis labelled by the spin \(j\)  and the magnetic moment \(m\)  running from \(-j\)  to \(+j\)  by integer steps.
The spin \(j_{e}\)  carried by an edge \(e\)  gives its quantized length in Planck unit, \(\ell _{e}=j_{e} \ell _{Planck}\) . A state in \({\mathcal {V}}_{j}\)  is then interpreted as a quantum 3-vector of length \(j\) . This is the key to the geometrical interpretation of the Ponzano-Regge model.

2-cells (faces):
Faces are assumed to have the topology of a two-dimensional disk with a \({\mathcal {S}}^{1}\)  boundary.
They are dressed with intertwiners, that is \(\mathrm {SU}(2)\) -invariant states living in the tensor product of the spins living on the edges forming the face's boundary:
\({\mathcal {I}}_{f}=\textrm {Inv}_{\mathrm {SU}(2)}\bigg {[}\bigotimes _{e\in \partial f}{\mathcal {V}}_{j_{e}}\bigg {]}\,.\) 
In the case of a triangle, consisting of three edges carrying spins \(j_{1},j_{2},j_{3}\) , this intertwiner state is one-dimensional if the spins satisfy the triangular inequalities and 0-dimensional in all other cases. When it is non-trivial, the unique intertwiner state is given by the Clebsh-Gordan coefficients, or equivalently the Wigner \(\lbrace 3j\rbrace \) -symbols, encoding the recouplings of the three spins.
In general, intertwiner states are interpreted as the quantum version of polygons.

3-cells (elementary 3D regions or bubbles):
Elementary 3-cells \(\sigma \)  have the topology of a 3-ball. Their boundary has the topology of a 2-sphere made of faces glued together along edges. We focus on the boundary of each 3-cell and think of them as bubbles. For each bubble, we usually introduce the dual graph \(\Gamma =(\partial \sigma )_{1}^{*}\) , defined formally as the 1-skeleton of the dual of the boundary 2-complex: each face is represented as a (dual) node and each edge is represented as a (dual) link linking two nodes. Each edge or link comes with its spin. Each face or node comes with its intertwiner state. Such graph with representations on its links and intertwiners at its nodes, \(\Gamma _{\lbrace j_{l},\iota _{n}\rbrace }\) , is called a spin network, as illustrated on fig.REF .
<FIGURE>We define the probability amplitude for the geometry of the 3-cell as the evaluation of its boundary spin network:
\({\mathcal {A}}_{\sigma }[\lbrace j_{e},\iota _{f}\rbrace _{e,f\in \partial \sigma }]=\text{Tr}_{\lbrace j_{e}\rbrace _{e\in \partial \sigma }} \, \bigotimes _{f\in \partial \sigma } \iota _{f}=\text{Tr}_{\lbrace j_{n}\rbrace _{n\in \Gamma }} \, \bigotimes _{n\in \Gamma } \iota _{n}={\mathcal {E}}_{\Gamma }[\lbrace j_{l},\iota _{n}\rbrace _{l,n\in \Gamma }]\,.\) 
The trace \(\text{Tr}\)  here is a slightly abusive notation. It means gluing the intertwiner states using the inner product on each edge, in the tensor product Hilbert space \(\bigotimes _{e\in \partial \sigma }{\mathcal {V}}_{j_{e}}\) .
In the case of a 3-simplex, or tetrahedron, the boundary graph consists of four nodes connected to each other through six links, as illustrated in fig.REF . The links carry six spins \(j_{1,..,6}\)  while the 3-valent nodes carry the corresponding unique intertwiner state numerically given by the Clebsh-Gordan coefficients. The resulting spin network evaluation is the celebrated Wigner's \(\lbrace 6j\rbrace \) -symbol.
In general, one could glue the intertwiner states by inserting \(\mathrm {SU}(2)\)  group elements \(g_{l}\)  (or even \(\mathrm {SL}(2,{\mathbb {C}})\)  group elements as in [3]}) along each link. This yields the spin network wave-function \(\psi _{\lbrace j_{l},\iota _{n}\rbrace }^{\Gamma }\in {\mathcal {C}}^{\infty }(\mathrm {SU}(2)^{\times E_{\sigma }})\)  where \(E_{\sigma }\)  counts the number of edges on the bubble boundary (or equivalently the number of links in the boundary graph \(\Gamma \) ). The spin network evaluation \({\mathcal {A}}_{\sigma }[\Gamma , \lbrace j_{l},\iota _{n}\rbrace ]\)  then truly is the evaluation of the spin network wave-function \(\psi _{\lbrace j_{l},I_{n}\rbrace }^{\Gamma }\)  on trivial group elements \(g_{l}=\mathbb {I}\)  in \(\mathrm {SU}(2)\) , reflecting the fact that physical states in pure 3D (quantum) gravity with vanishing cosmological constant have a flat curvature. The Ponzano-Regge model can indeed be understood as defining the projector onto the moduli space of flat \(\mathrm {SU}(2)\)  connections \(\textit {e.g.}\)  [4]}, [5]}.

